Homomorphism question.

I think I'm confused about notation perhaps.

let be a homomorphism

define for h,k in G

give a necessary and sufficient condition for it to be a homomorphism... prove why.

i know to show something is a homomorphism i have to show that , ie. that the binary operation's action is preserved in both groups. but i don't know how to here. for a=(m,n) and b=(i,j), is ab=(mi,nj)? that would give . How do I go about setting conditions to make these equal...? Seems to me they only would be if h,k were 0 or 1?

give a necessary and sufficient condition for it to be a homomorphism... prove why.

i know to show something is a homomorphism i have to show that , ie. that the binary operation's action is preserved in both groups. but i don't know how to here. for a=(m,n) and b=(i,j), is ab=(mi,nj)? that would give . How do I go about setting conditions to make these equal...? Seems to me they only would be if h,k were 0 or 1?

I think it must be , which sometimes is simply written = the free abelian group of rank 2.

It seems so, but the way you defined h doesn't make it clear: what are those two elements in G? Generators or what? Or h, k are only two arbitrary elements in G? If the latter then all it's requires is that h, k commute, though G itself can be non-abelian.

It seems so, but the way you defined h doesn't make it clear: what are those two elements in G? Generators or what? Or h, k are only two arbitrary elements in G? If the latter then all it's requires is that h, k commute, though G itself can be non-abelian.